number field

Start with the ordinary rational numbers and throw in one or more algebraic numbers — say the square root of 2, or a root of unity — then close up under the four arithmetic operations. You get a small, self-contained world of numbers that still sits inside the complex numbers but is much richer than the rationals. That world, when only finitely much new material was added, is a number field.

Precisely, a number field K is a field extension of the rationals Q of finite degree: [K : Q] is a finite number n, meaning K is an n-dimensional vector space over Q. Every number field is generated by a single algebraic number (the primitive element theorem), so K = Q(a) for some algebraic a whose minimal polynomial has degree n. The number field is the natural arena in which to do arithmetic beyond the ordinary integers.

Number fields carry a rich structure: a ring of integers playing the role Z plays in Q, a finite ideal class group measuring how badly unique factorization fails, a discriminant, and a collection of embeddings into the complex numbers. The whole apparatus of algebraic number theory is built to study these fields one prime at a time.